When a substance such as human tissue is subjected to a uniform magnetic field (polarizing field B0), the individual magnetic moments of the excited nuclei in the tissue attempt to align with this polarizing field, but precess about it in random order at their characteristic Larmor frequency. If the substance, or tissue, is subjected to a magnetic field (excitation field B1) that is in the x-y plane and that is near the Larmor frequency, the net aligned moment, Mz, may be rotated, or “tipped”, into the x-y plane to produce a net transverse magnetic moment Mt. A signal is emitted by the excited nuclei or “spins”, after the excitation signal B1 is terminated, and this signal may be sampled and processed to form an image.
When utilizing these “MR (magnetic resonance)” signals to produce images, magnetic field gradients (Gx, Gy and Gz) are employed. Typically, the region to be imaged is scanned by a sequence of measurement cycles, in which these gradients vary according to the particular localization method being used. The resulting set of received MR signals are digitized and processed to reconstruct the image.
The measurement cycle used to acquire each MR signal is performed under the direction of a pulse sequence produced by a pulse sequencer. Clinically available MRI systems store a library of such pulse sequences that can be prescribed to meet the needs of many different clinical applications. Research MRI systems include a library of clinically proven pulse sequences and they also enable the development of new pulse sequences.
The MR signals acquired with an MRI system are signal samples of the subject of the examination in Fourier space, or what is often referred to in the art as “k-space”. Each MR measurement cycle (referred to as TR in MRI literature), or pulse sequence, typically samples a portion of k-space along a sampling trajectory characteristic of that pulse sequence. Most pulse sequences sample k-space in a roster scan-like pattern sometimes referred to as a “spin-warp”, a “Fourier”, a “rectilinear” or a “Cartesian” scan. The spin-warp scan technique is discussed in [1]. The method, referred to as spin-warp imaging, employs a variable amplitude phase encoding magnetic field gradient pulse prior to the acquisition of MR spin-echo signals to phase encode spatial information in the direction of this gradient. In a two-dimensional implementation (2DFT), for example, spatial information is encoded in one direction by applying a phase encoding gradient (Gy) along that direction, and then a spin-echo signal is acquired in the presence of a readout magnetic field gradient (Gx) in a direction orthogonal to the phase encoding direction. The readout gradient present during the spin-echo acquisition encodes spatial information in the orthogonal direction. In a typical 2DFT pulse sequence, the magnitude of the phase encoding gradient pulse Gy is incremented (ΔGy) in the sequence of measurement cycles, or “views” that are acquired during the scan to produce a set of k-space MR data from which an entire image can be reconstructed.
The fundamental limitation imposed by magnetic field gradients on MRI data acquisition rates is that only a sequential acquisition of k-space samples along a particular sampling trajectory is possible. To circumvent this limitation, several approaches have been proposed and all under-sample k-space data. For instance in dynamic MRI [2]-[5], assumptions are made regarding the imaged object and/or temporal signal of interest to exploit data redundancy through k-space under-sampling. In compressed sensing (CS) [6]-[8], under-sampled data are projected onto a transform domain where its representation is sparse thus capturing signal information in a few samples. Parallel imaging (PMRI) [9]-[11] uses multiple receivers simultaneously to acquire under-sampled k-space, which is then restored to its full state by using receiver sensitivity profiles.
FIG. 1 shows an illustration 100 of an under-sampling technique applied on the k-space of an MRI image (k-space 1) formed by an array of sequential lines of data. Data that lies along four randomly chosen lines L1 to L4 in k-space 1 is shown as 104, 106, 108 and 110, respectively. In the under-sampling technique, only a portion of data (e.g. 104 and 108 that lie along lines L1 and L3) is acquired as 104′ and 108′ during the sampling process 102 to obtain an under-sampled k-space 1. During reconstruction, the original image is restored using the under-sampled k-space 1.
By under-sampling k-space, the volume of data acquired is sought to be minimized to reduce data acquisition time and thus accelerate a scan. However, the acceleration achievable with a technique is a function of its intrinsic properties. In CS, the limit is dictated by signal compressibility in the transform domain. In PMRI, the acceleration remains between 4 and 6 for two-dimensional (2D) Cartesian scans [12] that use coil arrays with as many as 128 channels [13]. Therefore, these methods have been combined in [14], [15] and also used with non-Cartesian sampling trajectories [16], [17], to increase acceleration.
One limitation common to all k-space under-sampling techniques is that signal to noise ratio (SNR) loss is unavoidable. Despite the advances, the challenges in increasing acceleration by minimizing acquisition SNR loss are nontrivial. This remains an area of interest [18] because of applications such as the evaluation of cardiac function that can always benefit from greater acceleration but are unable to do so because of SNR loss that can cause an unacceptable degradation in image quality.
Recently, a new method [19] for simultaneously acquiring distinct k-space data samples was introduced. During acquisition, the method uses a combination of RF pulses and gradients to overlap distinct k-space samples. Therefore, the data from the resultant acquisition consists of overlapped k-space samples and is referred to as “aliased k-space”. In this method [19], the “aliased k-space” acquisition technique is designed exclusively for dynamic MRI applications wherein the overlapped k-space samples are also “tagged” in time during the acquisition. This tagging process enables an unfolding of the overlapped k-space samples during reconstruction.
FIG. 2 shows an example 200 of an overlapping technique applied on an MRI data (k-space 1) formed by an array of sequential lines of data. Data that lie along four randomly chosen lines L1 to L4 in k-space 1 is shown as 204, 206, 208 and 210, respectively. In the overlapping technique illustrated in FIG. 2, one or more such lines of data are acquired in an aggregated manner. For example, all data along the lines L1 and L2 is simultaneously captured, while all data along the lines L3 and L4 is simultaneously captured. In this case of the four chosen lines L1 to L4, the data aggregation may result in the formation of two groups: one comprising data 204 and 206 that lies along adjacent lines of data L1 and L2, the other comprising data 204 and 206 that lies along adjacent lines of data L3 and L4. The resulting data (Overlapped k-space1) from the overlapping technique would then have a first line of data (204+206)′ that comprises data lying across L1 and L2; and a second line of data (208+210)′ that comprises data lying across L3 and L4. During reconstruction, the original image is obtained by unfolding this Overlapped k-space1.
Subsequently, an improvement on this technique was introduced [20] in which it was demonstrated that the unfolding of the overlapped k-space samples can also be achieved by using receiver sensitivity profiles. As a result, the technique introduced in reference [19] is no longer restricted to dynamic MRI technique and becomes applicable for all MRI scans.
A unique property of “aliased k-space” acquisitions is that depending on the manner in which the k-space samples are overlapped, it is possible to actually preserve, if not enhance, SNR (signal to noise ratio) during data acquisition while simultaneously accelerating the scan. A full description of the “aliased k-space” data acquisition, the restoration using receiver sensitivity profiles, and the related SNR properties has been described in detail in [20].
It is also relevant to note that while certain types of multi-slice PMRI acceleration techniques [21]-[22] can also be interpreted to be the overlap of k-space samples, this is an outcome of a process that overlaps different imaging slices onto each other. In these approaches, the overlapping of data samples only occurs along a spatial dimension (x, y or z). In contrast, in the “aliased k-space” acquisition approach described in [19]-[20], the overlapping samples originally lie along a phase-encoding axis within the k-space of a single slice or a single volume. Therefore, the acquisition approach in [19]-[20] brings about the ability to directly overlap k-space samples that lie along phase encoding axes in k-space, thus differentiating it from the approach described in [21]-[22]. Hence, it is now possible to perform “aliased k-space” acquisitions of a single slice or a single volume, as the case may be.
FIG. 3 shows an example of a combination of RF pulse 302 and gradient pulse 304, 306 and 308, designed for an “aliased k-space” acquisition that overlaps three blocks of k-space data. In FIG. 3, waveform 302 represents the three RF pulses, waveform 304 represents the phase encoding gradient blips (Gpe). “A” refers to amplitude of slice select gradient pulses (Gsl) 306. Waveform 308 represents readout gradient (Gr). In waveform 302, θ1, θ2 and θ3 refer to flip angles of the three RF pulses in 302. The resultant signal in an “aliased k-space” 2D acquisition using this combination of RF pulse 302 and gradient pulse 304, 306 and 308 is given by the following:S(t)=∫∫O(x,y){a1e−ik1y+a2e−ik2y+a3e−ik3y}e−kxxdxdy  [1]
In Equation [1], k=γ(Gpe1+Gpe2+Gpe3)ytp, k2=γ(Gpe2+Gpe3)ytp, k3=γ(Gpe3)ytp, tp is the duration of the phase encoding (PE) gradient blips, O(x,y) is a 2D axial slice, y is gyromagnetic ratio, a1, a2, and a3 are the amplitudes of the signal components that the RF pulses 302 in FIG. 3 generate. After discretization of spatial locations and receiver sensitivities, the acquisition process represented by Equation [1] can be re-written in matrix form as follows:
                              E                                    (                              n                ,                l                            )                        ,            ρ                          =                              [                                          ∑                                  m                  =                  1                                                  R                  k                                            ⁢                                                a                  m                                ⁢                                  e                                                            ik                      n                      m                                        ⁢                                          r                      ρ                                                                                            ]                    ⁢                                    C              l                        ⁡                          (                              r                ρ                            )                                                          [        2        ]            
In Equation [2], rρ denotes the pth voxel location, knm is the mth sample at the nth location in aliased k-space, Cl is the spatial sensitivity profile of the lth MRI receiver and am are the amplitudes of the signal components that are aliased to obtain the “aliased k-space”.
Accordingly, an “aliased k-space” acquisition is given by:Ev=d  [3]
In Equation [3], E is the encoding matrix, v is the desired image vector and d is the acquired aliased k-space data from all receivers. This encoding matrix E can employ either the image domain or the k-space equivalent of the receiver sensitivity profiles. Additional details on these signal acquisitions and the subsequent data restoration stages have been provided in [20].
Of all the data overlapping patterns that are applicable in an “Aliased k-space” acquisition, FIG. 4 shows an overview of a process 400 using the “block aliasing” approach described in [20]. In FIG. 4, k-space data 404 of the original image 402 is divided into 3 equal blocks A 406, B 408 and C 410, which are weighted with the amplitudes a1, a2 and a3 respectively and added to obtain the overlapped k-space data block 412. As discussed above, this “block aliasing” approach described in [20] possesses the property of preserving/enhancing SNR while simultaneously accelerating an MRI scan.
However, the primary challenge in restoring data that has been acquired through the “block aliasing” process 400 is that the encoding matrix E as defined above is poorly conditioned. This brings the following shortcomings:
1) Output image quality is extremely sensitive to errors in the estimates of the receiver spatial sensitivity profiles; and
2) The data restoration process needs a large number of receiver coils to be employed and the output image quality is dependent on receiver geometry.
In addition to these shortcomings, post reconstruction noise amplification is also high due to the poor condition of E. Therefore, an improvement on the condition of E for the block aliasing acquisition pattern is highly desirable in order to enhance the accuracy of and the quality of the output image from the data restoration process.
The present invention seeks to address the above shortcomings of “block aliasing” when restoring images acquired during MRI scans.